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Theorem imass1 6009
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 5918 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5848 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5602 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5602 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 3966 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3887  ran crn 5590  cres 5591  cima 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602
This theorem is referenced by:  predrelss  6240  vdwnnlem1  16696  dprdres  19631  imasnopn  22841  imasncld  22842  imasncls  22843  utoptop  23386  restutop  23389  ustuqtop3  23395  utopreg  23404  metustbl  23722  imadifxp  30940  esum2d  32061  eulerpartlemmf  32342  bj-imdirco  35361  brtrclfv2  41335  frege97d  41360  frege109d  41365  frege131d  41372  hess  41388  resimass  42784  setrecsss  46406
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